Perfect Square Formula: The formula for perfect squares is used to determine the squared value resulting from adding or subtracting two terms, (a ± b)². This formula is utilized for both algebraic computation and factorization purposes. When applied, it give the square of the sum or difference of two terms. The perfect square formula can be represented as:
(a ± b)² = (a² ± 2ab + b²)
Perfect Square Formula Solved Examples
Example 1:
Given the expression x
2
+12x+36, verify if it's a perfect square.
Solution: Rearranging the terms,
x
2
+12x+36=
x
2
+2×6×x+6
2
Utilizing the perfect square formula,
a
2
+2ab+b
2
=(a+b)
2
, the expression becomes(x+6)
2
.
Therefore, x
2
+12x+36 is a perfect square trinomial.
Example 2:
Simplify the expression 9a
2
−30ab+25b
2
using the perfect square formula.
Solution: The given expression is 9a
2
−30ab+25b
2
. Breaking it down into perfect square form, we get:
9a
2
−30ab+25b
2
=(3a)
2
−2×3a×5b+(5b)
2
Applying the perfect square formula, we get (3a−5b)
2
.
Therefore, 9a
2
−30ab+25b
2
=(3a−5b)
2
.
Example 3:
Find the square of the binomial 4x−3y using the perfect square formula.
Solution: Given a=4x and b=3y, the expression to be squared is (4x−3y)
2
. Using the perfect square formula, a
2
−2ab+b
2
: (4x)
2
−2×(4x)×(3y)+(3y)
2
This simplifies to 16x
2
−24xy+9y
2
.
Therefore, (4x−3y)
2
=16x
2
−24xy+9y
2
.
Example 4:
Determine if the expression x
2
+10x+25 is a perfect square.
Solution: The expression is x
2
+10x+25.
Rearranging it, we have x
2
+2×5×x+5
2
.
Using the perfect square formula (a+b)
2
=a
2
+2ab+b
2
, the expression simplifies to (x+5)
2
.
Hence, x
2
+10x+25 is a perfect square.
Example 5:
Find the value of (3a−2b)
2
.
Solution: Given a=3a and b=2b, we need to determine the square of the binomial (3a−2b)
2
.
Using the perfect square formula a
2
−2ab+b
2
:
(3a)
2
−2×(3a)×(2b)+(2b)
2
This simplifies to 9a
2
−12ab+4b
2
.
Therefore, (3a−2b)
2
=9a
2
−12ab+4b
2
.
Example 6:
Simplify the expression (x+7)
2
−49.
Solution: The expression is (x+7)
2
−49.
This can be written as (x+7)
2
−7
2
.
Using the formula a
2
−b
2
=(a+b)(a−b), this becomes (x+7+7)(x+7−7). Simplifying further, it's (x+14)(x), which can be written asx(x+14). Therefore, (x+7)
2
−49=x(x+14).
Example 7:
Evaluate the expression 16x
2
−40xy+25y
2
using the perfect square formula.
Solution: The given expression 16x
2
−40xy+25y
2
can be represented in perfect square form as follows: 16x
2
−40xy+25y
2
=(4x)
2
−2×4x×5y+(5y)
2
Applying the perfect square formula a
2
−2ab+b
2
=(a−b)
2
, it simplifies to (4x−5y)
2
.
Therefore, 16x
2
−40xy+25y
2
=(4x−5y)
2
.
Example 8
: Given the expression 2x
2
+12x+18, determine if it's a perfect square trinomial.
Solution: The expression is 2x
2
+12x+18.
Rearranging terms, we get 2x
2
+2×3×x+3
2
.
Using the perfect square formula(a+b)
2
=a
2
+2ab+b
2
, the expression simplifies to (x+3)
2
. Therefore, 2x
2
+12x+18 is a perfect square trinomial represented as (x+3)
2
.
These examples illustrate different scenarios of applying the perfect square formula to simplify expressions or determine whether they are perfect squares. The various examples showcased the utility of the perfect square formula in identifying perfect squares, simplifying expressions, and finding squared values of binomials. Understanding and applying this formula greatly aids in mathematical calculations and problem-solving within algebraic contexts.
The perfect square formula, (a±b)
2
=a
2
±2ab+b
2
, plays a fundamental role in algebraic computations and factorization. It assists in determining the squared value resulting from the addition or subtraction of two terms. By using this formula, expressions can be simplified or identified as perfect square trinomials.
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